Representation Theoretical Methods in the Theory of Special Function

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The proposed research involves computer explorations and the transfer of information connecting several areas of mathematics, most particularly Representation Theory, the Theory Special Functions and Combinatorics. These connections provide a powerful vehicle of discovery, since results and mechanisms which may be quite obvious in one area often translate into highly nontrivial and unexpected facts in one of the other areas. This type of activity is also highly suitable for training young researchers and providing them with the opportunity to discover the manner in which some research can be carried in our Computer Age. Combinatorial interpretations translate mathematical information, be it algebraic, analytical, logical or otherwise, into visual information. Under this setting, even students with limited background can be brought to experience the joy of discovery. For these reasons the PI, plans to continue the practice of bringing all current research efforts right into the classroom through graduate courses and seminars. In fact most of the work of the students and collaborators of the PI carried out under prior NSF support resulted from such classroom activities. The subfield of Algebraic Combinatorics, that is the focus of the research carried out under this award was created by the PI's representation theoretical approach to the (1988) conjectures by Macdonald concerning his now-famous symmetric polynomial basis. The 1990's joint work of the PI and Mark Haiman led to a conjectured formula for the Frobenius characteristic of the space ``Diagonal Harmonics'' in terms of the Macdonald polynomials. Early computer explorations by the PI and Haiman yielded data which revealed a surprisingly intimate connection of Diagonal Harmonics with ``Parking Functions'' (a colorful combinatorial structure created by computer scientists.) These discoveries led to the formulation by Haglund et al. of the Shuffle Conjecture which gives the Frobenius Characteristic of Diagonal Harmonics a beautiful explicit formula in terms of Parking Functions. Around the year 2000, Mark Haiman proved the original conjectures formulated jointly with the PI by Algebraic Geometrical tools. This brought the attention of the Algebraic Geometers to this field of investigation. This attention resulted in a truly surprising enrichment of the field with a variety of new tools and open problems. Very recently the PI and his collaborators, using these new findings succeeded in formulating an infinite family of ``Shuffle Conjectures'' connecting a whole Lie Algebra of Symmetric Function Operators to new ``Parking Function'' like objects. Under the support of this award the PI plans to use the vast collection of tools developed in two decades of efforts in this area to work on these problems in collaboration with his present PhD students Emily Leven, Yeonkyung Kim and Marino Romero. Preliminary results obtained by the PI and his present students have been very promising. In fact, recently Leven has succeeded in solving an infinite subfamily of these new conjectures. Historically, difficult problems have been the source of fundamental mathematical discoveries. Our particular area of investigation should be no exception in this respect.